# When did physics texts start to teach Kepler's $3/2$'s power law as a result of Newton's $1/r^2$ law of gravitation, rather than the other way around?

In modern physics textbooks, we teach Newton's laws of motion, then Newton's law of Universal Gravitation, and then Kepler's laws of planetary motion. Specifically, from the $1/r^2$ form of the gravitational force, and some other parts of Newton's laws, we can derive Kepler's 3rd law, that the period of motion of a planet is proportional to the $3/2$ power of its distance from the sun.

But historically, Kepler developed his laws before Newton wrote the Principia. Newton formulated his laws in the Principia, then (also in the Principia) derived the specific $1/r^2$ form of his gravitational law from the $3/2$ power form of Kepler's 3rd law.

My question is: when did physics texts and/or courses switch from the historical order of these two laws to the more recent (and possibly more pedagogical)? Was there a reason given at the time? The historical order was more inductive in its reasoning, while the modern presentation is more deductive in its reasoning.

One possibility I can think of is that we derive the $1/r^2$ form of Coulomb's law using Gauss's law and the fact that (macroscopic) space is 3-dimensional. That derivation carries over word-for-word to gravity. That becomes a very logical reason to say gravity should have the $1/r^2$ form once you know vector calculus. That might be a fruitful time period to look at.

• The first text to have derived Kepler's law from Newton's law is surely Principia itself. Do you have any evidence that texts after that kept teaching Kepler's law before Newton's? From a modern perspective that would seem rather strange, but I don't know much about physics pedagogy in the 17th and 18th centuries so I can't rule it out. – Logan M Oct 29 '14 at 3:30
• No. Principia takes Kepler's laws as given, then derives the 1/r^2 from of gravity from them. It does not derive Kepler's law from Newton's. I don't have any evidence about what happened after the publication of Principia; that's what I'm asking. – Colin McFaul Oct 29 '14 at 3:39
• Yes, it seems you're correct about that. Principia also seems to not be the earliest work to contain this derivation, as De motu corporum in gyrum predates it by 3 years. I don't know when the philosophical switch happened as to which is fundamental. – Logan M Oct 29 '14 at 4:13
• Very interesting question!! – Danu Oct 29 '14 at 8:11
• @LoganMaingi I can't see why you would find strange that some deduce a general law from an observation. Until Cavendish's experiment of 1798 I can't see any reason for not treating Newton's law as a result of Kepler's law. – VicAche Nov 1 '14 at 12:32

I am not familiar with physics textbooks of 18 century, so I do not know the answer to the question. However I want to add a comment which is too long for the comment window. (The system does not allow me to post a comment of this length.)

Principia actually contains both derivations. Of the inverse square law from Kepler laws and of the Kepler laws from the inverse square law. Kepler laws are empirical laws. While the law of gravitation is a theory. At the time when Principia was written, the equivalence of the law of gravitation to the Kepler laws was the crucial proof of the law of gravitation. Only later in 18 century other consequences of the law of gravitation were tested (the shape of Earth, for example, and most notably the theory of the Moon motion). Until the triumph of the Moon theory in the second part of 18 century, there were doubts about the inverse square law.

Why Kepler laws are taught nowadays as consequence of the Newton's law? I am not sure that this is the case in the astronomy books for the beginners. And elementary physics books do not contain actual derivation of the Kepler laws from the law of gravity. I was taught in high school both Kepler laws and the law of gravitation, and I was TOLD that Kepler's laws are a consequence, but no actual derivation was given.

It is not always convenient to follow the historical development of the subject in teaching. For example, we are not taught Ptolemy system, neither in astronomy nor in physics courses. And this is not because it is "wrong":-) It is not wrong. If you look to the modern Nautical almanac, it conforms more to the Ptolemy system than to the heliocentric one. And the actual computation of ephemerides uses their representation as trigonometric series, in complete conformity with "epicycles".

Yet another data point, the famous conversation between Newton and Halley a year and half before the Principia:

In 1684 Dr Halley came to visit him at Cambridge. After they had been some time together, the Dr asked him what he thought the curve would be that would be described by the planets supposing the force of attraction towards the sun to be reciprocal to the square of their distance from it. Sir Isaac replied immediately that it would be an ellipse. The Doctor, struck with joy and amazement, asked him how he knew it. Why, saith he, I have calculated it. Whereupon Dr Halley asked him for his calculation without any farther delay. Sir Isaac looked among his papers but could not find it, but he promised him to renew it and then to send it him...
• While interesting, this does nothing to answer the question as asked: when did the switch occur? (Thus, some time after Principia.) – Michael Weiss Dec 10 '14 at 15:03